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Problem 5. Define a binary relation ~ on N² by for all n, m, n', m' E N. \left(n_{0}, n_{1}\right) \sim\left(m_{0}, m_{1}\right) \Longleftrightarrow n_{0}+m_{1}=m_{0}+n_{1} (b) Define a function f :

N² x N² → N² by f (nо, n1), (mо, m1)) = (по + mо,n1+ m1). Prove that f respects the equivalence relation ~. (c) Let f be as in part (b) and let f : (N/ ~)² –→ (N/ ~) be the function induced by f, f((no, n1)]~,[(mo, m1)]~) = [f((no,n1), (mo, m1))]~- \text { Prove that for every }\left[\left(n_{0}, n_{1}\right]_{n} \in \mathbb{N}^{2} / \sim, \text { there exists }\left[\left(m_{0}, m_{1}\right)\right]_{\sim}\right. \text { such that } \tilde{f}\left(\left[\left(n_{0}, n_{1}\right)\right]_{\sim},\left[\left(m_{0}, m_{1}\right)\right]_{\sim}\right)=[(0,0)]_{\sim} (a) Prove that ~ is an equivalence relation on N².

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